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A297890 Decimal expansion of lim_{k->infinity} (A291481(k)^(1/k)).

%I A297890 #10 Jan 10 2018 03:34:50
%S A297890 2,2,3,6,1,5,1,3,7,7,4,6,8,7,4,9,0,3,9,6,6,2,7,7,8,4,0,6,6,1,2,0,4,4,
%T A297890 0,7,3,9,1,1,4,6,4,9,9,2,4,0,2,3,9,5,7,5,4,6,3,2,1,2,5,5,5,2,6,6,5,7,
%U A297890 9,7,4,3,1,9,3,6,4,0,2,8,6,2,6,9,8,1,0
%N A297890 Decimal expansion of lim_{k->infinity} (A291481(k)^(1/k)).
%C A297890 Equals 2^(2 - d) * 3^(d - 1), where d = lim_{k->infinity} (1/k)*Sum_{i=1..k} A293630(i) = 1.275261... (see A296564).
%C A297890 See comments from _Jon E. Schoenfield_ on A296564 for explanation of PARI program.
%C A297890 Is this number transcendental?
%H A297890 Iain Fox, <a href="/A297890/b297890.txt">Table of n, a(n) for n = 1..20000</a>
%e A297890 Equals 2.2361513774687490396627784066120440739114649924...
%e A297890 Values evaluated with A291481(k):
%e A297890   k = 1: 4^(1/1)                 = 4
%e A297890   k = 2: 7^(1/2)                 = 2.645751311064590590...
%e A297890   k = 3: 13^(1/3)                = 2.351334687720757489...
%e A297890   k = 4: 37^(1/4)                = 2.466325714559660444...
%e A297890   k = 5: 73^(1/5)                = 2.358655818240735626...
%e A297890   k = 6: 145^(1/6)               = 2.292070651723655173...
%e A297890   ...
%e A297890   k = infinity: A291481(k)^(1/k) = 2.236151377468749039...
%o A297890 (PARI) gen(build) = {
%o A297890 my(S = [1, 2], n = 2, t = 3, L, nPrev, E);
%o A297890 for(j = 1, build, L = S[#S]; n = n*(1+L)-L; t = t*(1+L)-L^2; nPrev = #S; for(r = 1, L, for(i = 1, nPrev-1, S = concat(S, S[i]))));
%o A297890 E = S;
%o A297890 for(j = build + 1, build + #E, L = E[#E+1-(j-build)]; n = n*(1+L)-L; t = t*(1+L)-L^2);
%o A297890 return(2^(2 - t/n)*3^(t/n - 1))
%o A297890 } \\ (gradually increase build to get more precise answers)
%Y A297890 Cf. A291481, A293630, A296564.
%K A297890 cons,nonn
%O A297890 1,1
%A A297890 _Iain Fox_, Jan 07 2018